1.4 Networks, topologies and entanglements

Lucia Carlucci, Gianfranco Ciani and Davide M. Proserpio

Dipartimento di Chimica Strutturale e Stereochimica Inorganica,

Via G. Venezian 21, 20133 Milano, Italy.

1.4.1 Introduction

Making crystals by design requires both a detailed knowledge of the nature of the interactions

involving the selected building blocks and also some general concepts concerning the types of the

supramolecular extended arrays that could be generated by self-assembly, i.e. the possible target

networks. The presence of suitable functional groups in an organic building block or the free

coordination sites in a metal-based centre, all having well defined directions for interactions, can

allow for some previsions about the resulting architecture, thus representing the fundamentals of the

modern area of crystal engineering of networks. These points are treated in great detail in other

parts of this book. However, in spite of this rather optimistic incipit, we are presently faced with

many problems in the field and we are still far from being able to deliberately build networks with

desired properties. A major difficulty arises from the range of possible different structures that can

form in the self-assembly process, due to the occurrence of many, often unpredictable and subtle,

concurrent factors.

Some advantage can derive from the analysis of related structures reported in the literature,

searching for prototypical models that can fit the specific problem. The increasing number of

networked species reported in the last years offers a rich variety of new structural types that

continuously amplify our knowledge of these organic or metal-organic extended systems.

From the structural point of view a great attention has been devoted to the rationalization and

classification of the different topological types of single extended motifs [1] and to the

categorization of the frequently encountered interpenetrating [2] and entangled [3] species. Indeed,

the "network approach" or topological approach to crystal chemistry [4,5] is an important

achievement in this area, as a useful tool for the analysis of network structures, in that it simplifies

complex species to schematised nets, in order to make comparisons and identify packing trends that

may help in the rational design of functional materials.

In what follows we will describe some approaches to the rationalization of these supramolecular

systems proceeding in order of increasing complexity. We will deal first with the simplification of

crystal structures, in order to carry out a topological analysis of the individual idealized networks.

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Problems concerning the unambiguous assignment of the topology are presented, and topological

approaches towards making crystals are described. Various types of entanglements of single motifs,

including interpenetrated, polycatenated, Borromean, polythreaded and other networked arrays,

will be then considered and classified.

1.4.2 Rationalization and simplification of the extended structures

A correct analysis of the crystal structures is fundamental in order to avoid misinterpretations, that

can easily occur, about the network topology and, when present, the type and extent of the

entanglement of distinct motifs. The rationalization of these, often complicated, structures implies a

sequence of steps leading to the basic nets of linked nodes.

The steps, in general, can be summarized as follows: i) simplification process, ii) identification and

separation of the individual motifs, iii) topological analysis of these motifs, and iv) topological

analysis of the whole entanglement.

The simplification phase, previously exemplified by some authors [6], consists in the rather obvious

operation of removing all the unnecessary elements that have no topological relevance, thus leaving

only the essentials, represented by nodes and links (vertices and edges, respectively, in graph

theory [7]). For instance, polyatomic nodes (like metal clusters or polyfunctional ligands) can be

replaced by their baricentres.

The simplified model must be effectively representative of the connectivity of the real network. In

some cases, however, this step leads to structural descriptions that are rather subjective in the

selection of both the nodes and the connections (links) joining the nodes. Alternative

rationalizations can be accomplished, that result in motifs of different topology, depending upon the

choice of the chemical or/and mechanical bonds and the nodes of the architecture (see Figure 1.4.1)

[8]. These "alternative" motifs, however, are not something unnatural; they reflect the possibility of

consideration of a crystal from different points of view or at various levels of structure organization.

FIGURE 1.4.1

In coordination networks concurrent chemical forces are often operative, including coordinative

bonds, hydrogen bond bridges, secondary bonds involving the anions, metal-metal interactions, π-

interactions and others. Thus we are faced with the problem of deciding when a link is a

topologically significant link. A current useful practice consists in assuming a level within this

hierarchy of forces to be applied in the description of the net. For instance, we have previously

observed [3] that in [Zn(4,4’-bipy)2(H2O)2](SiF6) [11] the Zn(4,4’-bipy)2 coordination sublattice is a

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3D polycatenated array of two sets of inclined 2D square layers, but including also H2O-SiF6-H2O

hydrogen bond bridges that link the two different sets results in a more complex 3D single net

having 4- and 8-connected nodes, with the peculiar feature of self-catenation (see later). Thus the

insertion in a model of additional weaker interactions can have drastic effects, like changing the

topology of the individual motifs, the type of entanglement or also the dimensionality of the whole

array.

FIGURE 1.4.2

Mechanical bonds, like those shown in Figure 1.4.2, are responsible for the formation of the

majority of the known extended periodic entanglements. A debatable point concerns the difficult

distinction between rotaxanes and pseudo-rotaxanes. At the molecular level rotaxanes are

characterized by the presence of bulky stoppers on the rods, that must inhibit dethreading, while

pseudo-rotaxanes can be separated by chromatography or at high temperatures. Also some

rotaxanes, however, can have low energy barrier to dethreading, so that the difference with respect

to pseudo-rotaxanes remains a delicate problem and the boundary is not well-defined [12]. The

extended polymeric analogues of these species present problems of similar nature; moreover, for

them no experiment can be envisaged to differentiate the two situations.

We could, therefore, conclude that establishing the topology of these species is not only a formal

process but rather an interplay between chemistry and mathematical theory. The process of

simplification or abstraction that represents the passage from real objects to their idealized models

needs chemical considerations. After simplification the problem remains exclusively based on a

mathematical ground, and the successive process consists in the identification and classification of

the topology of the individual motifs contributing to the entanglement. Their separation can be

accomplished in a computer-aided process, whose output is a set of distinct "coloured" nets

[13,14,15].

1.4.3 Topological classification of networks

1.4.3.1 Nomenclature for single nets: Schläfli and Vertex Symbols

The topological classification of an individual extended motif can be accomplished according to

generally accepted criteria used in a variety of structural contexts, that are here described. Before of

this, however, we want briefly mention the pioneering work of the researchers that have introduced

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the basic concepts for the rationalization, enumeration and classification of the extended crystal

structures, using different approaches.

A fundamental contribution is represented by a famous series of articles and books on crystal

chemistry published many years ago by A. F. Wells [4], who analysed and classified a great number

of nets. He emphasized the importance to describe a crystal structure in terms of its basic topology:

such a description not only provides a simple and elegant way of representing the structures but also

evidences relations between structures that are not always apparent from the conventional

descriptions. Wells introduced a method for the systematic generation of 3D arrays from 2D nets

and described also many hypothetical motifs that were successively discovered within the realm of

coordination polymers or of other extended systems. His results included a list of many simple nets

described with only one kind of node (uninodal) or with two nodes of different connectivity (mainly

binodal 3,4-connected).

The studies by J.V. Smith focused on the classification of zeolites and related materials with

tetrahedral nodes of the TO4 type [16]. He described a prolific number of frameworks

that were discovered mostly by hand treatment, using systematic methods. Such methods include

permutation of up-down linkages between identifiable two-dimensional layers in zeolites, as well as

permutations of linkages between identifiable polyhedra.

An empirical search by O’Keeffe [17] for uninodal four-connected nets resulted in the recognition

of 168 distinct types (many of which unseen by Wells).

A thousand of uninodal nets were described over the years by Fischer, Koch and Sowa in their

enumeration of homogeneous sphere packings, exploring systematically all crystal systems [18].

M. M. J. Treacy and coworkers have collected a hypothetical zeolite database, using a method that

enumerates all possible 4-connected nets within each crystallographic space group given the

number of unique tetrahedral vertices. The database contains 1010 graphs and is available at the

website www.hypotheticalzeolites.net [19].

The different network topologies can be recognized following the graph theoretical approach. We

can assign to the simplified 2D or 3D crystal structure a net, that is an infinite graph (or a periodic

connected graph) described with translational symmetry in exactly two or three independent

directions (2-periodic or 3-periodic nets) [20] and represented by nodes and links (vertices and

edges, following the nomenclature of graph theory [7,21]). Figure 1.4.3 illustrates some common

net topologies.

FIGURE 1.4.3

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In the analysis of the topology of such nets we can look for a classification scheme that will allow

to uniquely assign equal nets (up to isomorphism) derived from totally different crystal structures.

Remember that the topology is not influenced by the metrical properties of the structure (angles,

distances), so that a 4-connected diamondoid net is such even if highly distorted (the geometry

around the nodes could be far from tetrahedral), and, obviously, also not dependent on the chemical

nature of each node/vertex.

To make it easy we suggest here to follow the lower-case three-letter code nomenclature for nets

proposed by O'Keeffe and coworkers [22,23] in the attempt to overcome a certain confusion in the

field, due to the fact that some networks have been described with many names and symbols. For

example the srs net observed for the Si atoms in the SrSi2 structure (see Figure 1.4.3) has been

called (10,3)-a, Laves net, Y*, 3/10/c1, or labyrinth graph of the gyroid surface. The suggested

nomenclature is designed to parallel the widely accepted upper-case three-letter codes used for

zeolite frameworks [24]. Moreover, more than 1300 nets with these symbols have been collected in

a very useful list described in detail at the RCSR (Reticular Chemistry Structure Resource) website

of O’Keeffe group (http://reticularchemistry.net/RCSR/ [25]).

For most of the nets of fundamental importance in crystal chemistry, there are only a few different

kinds of nodes (by "kind of nodes" we mean a set of vertices related by symmetry operations,

including translations) and the assignment of the local topology for 2D or 3D nets is based on the

analysis of the "circuits" (or cycles) at the angles of a node. A circuit is a closed path beginning and

ending at each node, characterized by a size equal to the number of edges comprised in the path (3-

circuit, 4-circuit and so on). Any two edges at a node define an angle, and with n-connected nodes

[26] there are [n(n-1)/2] such angles in a 3D net (but only n in a 2D simple layer [27]), for each of

which we can find a large number of different circuits of different sizes.

A n-connected node can be identified by a Schläfli Symbol (short symbol or point symbol)

of the form Aa.Bb.Cc..., in which A < B < C < ... and a + b + c ... = n(n–1)/2, that represents the sizes

(A, B, C...) and numbers (a, b, c...) of the "shortest circuits" contained at each angle. Thus for the 4-

connected diamond net (dia), where all the shortest circuits are 6-cycles, the symbol is 66. For the

2D honeycomb and square-grid nets and for the 3D net of the 6-connected primitive cubic lattice

(pcu), all uninodal (i.e. containing a single type of node), the Schläfli Symbols are 63 [or (6,3)], 44

[or (4,4)] and 412.63, respectively.

For multinodal nets the Schläfli Symbols are grouped according to the multiplicity of the node in

the unit cell; for example the net formed by the Si atoms in the silica form moganite (mog) is

described as the 4-connected binodal net (4.64.8)2(42.62.82) because there are two crystallographic

distinct nodes with multiplicity 2:1.

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A special type of point symbol is assigned to uniform nets, i.e. nets in which the shortest circuits at

each angle are all equal in size. It follows that the Schläfli Symbol for each node in a n-connected

uniform net with circuits of size A is An(n-1)/2. Examples include the series of 3-connected 103 nets

(srs, ths, bto, utp and others), the 4-connected 66 nets (dia, lon, gsi, lcs and others) and also the

binodal 3,4-connected (83)4(86)3 nets of C3N4 (ctn) and Pt3O4 (pto).

The "point symbol" was extensively used in a slightly different typographic form by Wells [4], who

was aware that this classification was far from being sufficient to unambiguously identify a net. In

fact he added letters to the point symbol for assigning different topologies, as in the case of the well

known 3-connected nets of the Si atoms in the SrSi2 and ThSi2 structures (srs and ths with the same

Schläfli Symbol 103), called (10,3)-a and (10,3)-b, respectively. Another case of ambiguity is given

by the NbO net (nbo) and the arrangement of Si atoms in quartz (qtz), that have the same Schläfli

Symbol 64.82.

Given this inadequacy other descriptors are needed. To differentiate nets with the same Schläfli

Symbol a modified version of it, the Vertex Symbol (VS, or long symbol introduced by O'Keeffe

[1]) appears more useful. In a VS the size of the shortest rings at each angle is given with a

subscript to denote the number of such rings. Solid state chemists define a ring as a circuit that has

the property that there is no shorter path ("short cut") between any two vertices on the circuit than

the shortest one that is part of the circuit (see Figura 1.4.4) [28].

FIGURE 1.4.4

Thus in the 3-connected srs 3D net there are five 10-rings at each angle and the VS is 105.105.105,

and for the ths net, with two, four and four 10-rings at each angle, the VS is 102.104.104 (remember

that for both nets the Schläfli Symbol is 103).

In the case of 4-connected nets there are six angles at a node. The diamond net (dia) with two 6-

rings at each angle has VS = 62.62.62.62.62.62. In 4-connected nets only, the angles are grouped into

three pairs of opposite angles, and subject to that constraint, the smallest numbers are given first.

The feldspar structure (fel) contains a 4-connected net with two kinds of vertices both having the

same Schläfli Symbol 42638, but different Vertex Symbols, 4.6.4.6.82.1010 and 4.62.4.8.6.62. Here

some 6- and 8-circuits have short-cuts and become 8- and 10-rings in the VS. Analogously we can

now distinguish NbO, with VS(nbo) = 62.62.62.62.82.82, from quartz, with VS(qtz) = 6.6.62.62.87.87.

The Vertex Symbol of PtS (cooperite pts), that contains "square-planar" Pt and "tetrahedral" S

centres with the same Schläfli Symbol 42.84, allows also to differentiate the two nodes: Pt

(4.4.82.82.88.88) and S (4.4.87.87.87.87). In this case the multiplicity of the 8-rings makes the

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difference. More details on these symbols are given in references [1,21,29]. They are used also in

the Atlas of Zeolite Framework Types [24a,b].

Not all the angles of a 4-connected net necessarily possess rings. When there is not a ring an asterisk

is inserted in the VS. Thus for the 4–connected CdSO4 net (the net of the Cd and S atoms, with the –

O– links considered as edges, cds, Schläfli Symbol 658) the Vertex Symbol is 6.6.6.6.62.* .

Vertex Symbols can be computed for higher connectivities, but since the number of angles

increases as the square of the coordination number, they soon become cumbersome; the Vertex

Symbol for the 6-coordinated net of the primitive cubic lattice (pcu) is 4.4.4.4.4.4.4.4.4.4.4.4.*.*.*,

and a 12-coordinated net has (12×11)/2 = 66 angles.

Also the Vertex Symbols, however, are unable to fully characterize nets. For example diamond and

lonsdaleite (hexagonal diamond, lon) have the same VS = 62.62.62.62.62.62 and, therefore, other

descriptors must be considered.

It is easy to introduce the "Coordination Sequence" CS(k) [1] of a node, defined as a sequence of

numbers (n1, n2, n3, …, nk, …) in which the kth term is the number of nodes in "shell" k that are

connected to nodes in "shell" k – 1. Shell 0 [CS(0)] consists of a single node and the number of

nodes in the first shell [CS(1)] is the connectivity of the node (or more rigorously the coordination

number of the vertex [26]). Another equivalent definition states that CS(k) gives the number of

"topological neighbours" as the number of nodes that are k edges/links away from a given node. It

follows that a kth neighbour of a node is one for which the shortest path to that node consists of k

edges. Though there is no limit for k the CS is usually computed up to k = 10. With this new

descriptor we can now distinguish dia (CS= 4,12,24,42..) from lon (CS= 4,12,25,44...). The

Coordination Sequences differ already at the third-neighbour, see Figure 1.4.5. We must say that CS

alone is not sufficient, also because there are nets with the same CS but different VS (for example

the zeolites Linde A lta and Rho rho).

FIGURE 1.4.5

For many years the structural analysis for assigning VS and CS was carried out by hand for simple

nets but nowadays it can be performed computationally with the use of available computer

programs like TOPOS [14,15,30]. This program gives VS and CS for each (crystallographically

independent) node. If all nodes result equal the net is classified as uninodal, otherwise binodal,

trinodal, ... n-nodal. A uninodal net is one in which all the nodes have the same VS and CS, i.e. are

congruent. In its maximum symmetry form there is only one node in the asymmetric unit and all

nodes will be related by symmetry operations (i.e. are equivalent). In real crystal structures, on the

other hand, it is common to find low symmetry embeddings of uninodal nets (an embedding of a

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graph is a realization of the graph in Euclidean space with a given geometrical description, atomic

coordinates, distances and angles) where there are crystallographic distinct nodes but all have the

same VS and CS, and therefore we can describe the topology of the derived net as that of an

uninodal net (see Figure 1.4.6).

Then the results can be compared with a list of known nets (e.g. more than 1300 nets described in

RCSR and the zeolite databases are stored in TOPOS) to find (if any) the topological type, with its

given name and ideal most symmetrical structure.

FIGURE 1.4.6

This pair of descriptors (Vertex Symbol and Coordination Sequence for all nodes) appears to be

unique for a particular net topology, i.e. they can be effectively employed to distinguish different

framework types unambiguously. This statement has not been proved rigorously but long

experience with databases has shown that, at present, only few theoretical different nets with the

same VS and CS descriptors are known, that can still be differentiated if all the rings (not only the

shortest) are computed. All these considerations belong to the context of graph theory: we can

always identify two nets as isomorphic ignoring edge crossings. Two different embeddings of a net

that can be deformed into each other only with edge crossings are called non-"ambient isotopic"

[21,31] and therefore cannot be distinguished in graph theory. In recent years, indeed, some

examples of non-"ambient isotopic" nets have been identified (see Figure 1.4.7) [3,32,33,34] for

which new topological descriptors are needed, taken outside the graph theory e.g. in the context of

knot theory [31].

FIGURE 1.4.7

Isomorphic nets can be rigorously recognized with the SYSTRE program by Delgado-Friedrichs

[35], that can establish the "equilibrium" configuration of nets and assign topological unique

coordinates to the vertices. By this we mean that each vertex has coordinates that are the average of

the coordinates of its neighbours. Once an origin is chosen, barycentric coordinates are unique for a

given choice of basis vectors. Such a placement is valuable in determining properties such as

combinatorial symmetry. "Collisions" occur when two vertices have the same barycentric

coordinates, but nets of such kind are still very rare in crystal chemistry (for further discussion of

"collisions" see ref. 35a). Thus, for nets without collisions, SYSTRE solves the graph isomorphism

problem: it unambiguously determines whether two nets have the same or different topology.

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It has been suggested [5,25,36] that a limited number of "common" topologies should be sufficient

to represent the large majority of organic, inorganic and metal-organic (MOFs) networks, thus

furnishing a useful "network approach" to the crystal engineering of these complex systems. This

important feature is true not only for single networks but also for interpenetrating 3D architectures.

The distribution of the topologies found for the 482 interpenetrated structures collected in CSD and

ICSD databases shows that the most common nets are by far the dia nets (ca. 42%), followed by

pcu nets (ca. 18%) and by srs nets (ca. 7%) [14,15].

1.4.3.2 Tiling theory and topological approaches towards making crystals

The current growing interest for network topologies has lead to investigate in novel directions,

especially for enumeration purposes, but also for crystal engineering. In the search of a rational

design of coordination polymers/MOFs it appeared evident the importance of the concept of

Secondary Building units (SBU) that was used for some times to describe conceptual fragments of

zeolites. When polytopic units are copolymerized with metal ions, it is useful to recognize linked

cluster entities in the assembled solid, each such cluster being considered a SBU. The successful

design of rigid frameworks based on SBUs is the rationale for the "Scale Chemistry" [37] and the

"Reticular Chemistry" [36b].

The theoretical background for the reticular synthesis arises looking at nets with a different

mathematical approach, i.e. moving from the periodic graph description to the tile description of

nets (based on combinatorial tiling theory) [38]. The tiles are generalized polyhedra (cages) which

generate the entire structure when packed together. For each periodic net there are many possible

tilings but a unique type called "natural" tiling [23a,23b,39]. The cube with eight vertices and six

faces formed by four-membered rings is the natural tile of the primitive cubic net (pcu) and is

symbolized as [46]. The adamantane unit with ten vertices is the natural tile of the diamond (dia)

net. Its four faces are six-membered rings and hence the symbol is [64]. The natural tiling of the

lonsdaleite net (lon) has equal numbers of two tiles that are [63] and [65], as shown in Figure 1.4.8.

This approach classifies the nets according to the "transitivity" of the structure: if there are p kinds

of vertices, q kinds of edges (links), r kinds of rings (faces of tiles) and s kinds of tiles, the

transitivity is [pqrs]. The five structures srs, nbo, dia, pcu, bcu whose natural tilings have

transitivity [1111] are called "regular". Their importance in reticular synthesis is comparable to that

of the five platonic solids in other areas of chemistry. There is only one structure with transitivity

[1112] called "quasiregular": the face-centered cubic net (fcu). The next class includes the 14

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"semiregular" nets with transitivity [11rs] [23b]. All together these twenty nets are the most likely

to form with one kind of SBU joined by one kind of link [40].

FIGURE 1.4.8

The field of infinite periodic minimal surfaces (IPMS), that was introduced few decades ago for the

analysis of the topology of crystal structures [41], is a different approach to the analysis of nets:

many common nets are related to the known intersection-free IPMS [42]. The IPMS studies has

produced also a systematic enumeration of nets that has been recently proposed by the EPINET

project (Euclidean Patterns in Non-Euclidean Tilings, see http://epinet.anu.edu.au): instead of

working directly in three dimensions, the intrinsic hyperbolic geometry of IPMS is used to map 2D

hyperbolic patterns into 3D Euclidean space [43].

The interested reader can find more information on the topology of networks in a recent review,

with more details on what do we now know about nets [33].

1.4.3.3 Self-catenated networks

In the topological analysis of a network one can encounter the unusual phenomenon of self-

catenation or self-penetration. Self-catenated nets are single nets that exhibit the peculiar feature of

containing rings through which pass other components of the same network. In more detail, we

must refer to the topological classification of nets, represented by the Vertex Symbol, assuming that

if one of the "shortest rings" is catenated by other "shortest rings" of the same net we can speak of a

true case of self-catenation.

Different terms have been employed for this type of self-entangled networks. Batten and Robson

[44] have underlined the relationship of these species with molecular knots (see Figure 1.4.2) but

they have preferred to classify them as self-catenating or self-penetrating nets. Champness,

Hubberstey, Schröder and coworkers have used the term "polyknotted coordination polymers" [45].

Also the term "intrapenetrating networks" has been employed [46]. Here we prefer to adopt the

term self-catenation since the catenation happens within the net itself and there are no different nets

involved. This unusual topological feature was previously observed also in simple

mineral/inorganic frameworks, as in coesite (coe), a high-pressure polymorph of silica [described

as 4-connected binodal net, with Schläfli Symbol (42.63.8)(42.6.82.9)], recognized by O’Keeffe to

be "unique among nets found in nature in that it contains 8-circuits that are linked as in a chain"

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(see Figure 1.4.9 middle) [47]. Later it was also recognized that two structures of ice, considering

hydrogen bonding, are self-catenated: Ice IV (icf) and Ice XII (itv) [48].

FIGURE 1.4.9

Probably the first examples of 3D self-catenated frameworks within coordination polymers were

described by Robson and coworkers [49]. In the species [Cd(CN)2L] [L = pyrazine or 1,4-bis(4-

pyridyl)butadiene] the Cd(CN)2 sublattices form 44 undulated layers that are joined in a criss-cross

fashion by the L spacers. Catenated hexagonal chair-like circuits are observed in the interlayer

connections (see Figure 1.4.9, left). The Schläfli Symbols of these 6-connected nets are 48.66.8

(rob) for L = pyrazine, and 44.610.8 for L = 1,4-bis(4-pyridyl)butadiene, respectively, so that the

requirements for self-catenation are fulfilled, since the catenated 6-rings are shortest rings [50].

Perhaps one of the most fascinating examples is represented by [Ni(tpt)(NO3)2] (tpt = tri-4-pyridyl-

1,3,5-triazine) [51]. This contains a remarkable 3-connected chiral uniform net of 123 topology

(twt VS=124.127.127) (see Figure 1.4.9, right) that was enumerated by Wells [4b] but never found

previously in real species. Within uninodal 3-connected nets this is special in that it contains

relatively "large" shortest rings (12-gons), the largest n-gons considered by Wells.

An intriguing feature of coordination network chemistry is its ability to produce rare and even only

hypothetical structural motifs, given the correct building blocks. Moreover the models are

reproduced on a larger scale. This is, inter alia, the case of [Ag(2-ethpyz)2](SbF6) (2-ethpyz = 2-

ethylpyrazine) that shows the coe topology [52].

Some coordination polymers exhibit notable structures containing independent polycatenated

motifs (vide infra) that are joined by bridging counterions or via supramolecular weak interactions,

thus resulting in an unique self-catenated net. A beautiful example of this class has been observed in

[Cd2(4,4’-pytz)3(µ-NO3)(NO3)3](MeOH) [4,4’-pytz = 3,6-bis(pyridin-4-yl)-1,2,4,5-tetrazine [45].

The structure consists of molecular ladders that give inclined catenation forming a 3D array. The

ladders of the two sets are cross-linked by µ-NO3- anions bridging the Cd(II) centers, resulting in

the formation of a single 3D polymer, that the authors describe as a "polyknot". It should be now

obvious that crossing of catenated layers (vide infra) results in a single self-catenated net [53].

1.4.4 Entangled systems

1.4.4.1 Types of entanglements

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Many of the intricate organic and metal-organic networks reported in the literature are particularly

intriguing because of the presence of independent motifs entangled together in different ways. After

the topological classification of the individual nets we must pass to a higher level of complexity, i.e.

to the analysis of the "topology of entanglements".

Entangled systems are extended arrays more complex than their constituents, that are comprised of

individual motifs forming, via interlocking or interweaving, periodic architectures infinite in at

least one dimension. Simple interdigitanion is not considered here. As previously stated, most of the

entangled arrays can be considered regularly repeated infinite versions of finite molecular motifs

like catenanes, rotaxanes and pseudo-rotaxanes.

Such molecular species are the subject of many topological investigations [9,10], especially

concerning the classification of isomers within complex organic molecules [54]. Thus, in treating

the complexity of entangled arrays, we will try to use concepts derived both from crystal chemistry,

that has classified inorganic nets, and from the mathematical theory of knots and links. Before

investigating the different structural classes we will examine some general preliminary aspects

concerning the topology of entanglements.

The most common type of entanglement is represented by the numerous family of "interpenetrating

networks" [2]. Interpenetration can be ascribed to the presence of large free voids in a single

network, though it was shown that this phenomenon does not necessarily prevent the formation of

open porous materials [55]. Wells introduced the theme of interpenetrating nets (identical or of two

or more kinds) by stating that they "cannot be separated without breaking links" [4b]. Few

examples were known at that time, including neptunite (2-fold interpenetrated ths), cuprite Cu2O

(one of the first crystal structures determined, 2-fold dia)), β-quinol (H-bonded 2-fold hexagonal

decorated primitive cubic pcu-h 6.102) , [Cu(adiponitrile)2](NO3), (6-fold dia) and some others.

Since then a great number of structural reports have appeared on interpenetrating nets, sustained

both by coordinative or by hydrogen bonds. The most important and comprehensive contributions

in the area can be found in a review by Batten and Robson [2] and in the successive highlights by

Batten [48a, 56]. According to these authors interpenetrating structures, that "can be disentangled

only by breaking internal connections", are characterized by the presence of infinite structurally

regular motifs that must contain rings "through which independent components are inextricably

entangled" [2].

However, within structures that are consistent with all the conditions described above, a distinct

subclass can be recognized, namely polycatenated nets. "Polycatenanes" have the peculiar features

that all the constituent motifs have lower dimensionality than that of the resulting entangled

architectures and that each individual motif is catenated only with the surrounding ones but not with

12

all the others, like a single ring of a chain. In principle, also 0D (finite) motifs could be comprised

within the possible component motifs. A comparison of the main aspects of polycatenation vs.

interpenetration is given in Figure 1.4.10. In between these two classes additional intricate

intermediate situations are also possible, like some recently found examples of interpenetrating 3D

plus 2D [57] and 3D plus 1D frameworks [58]

FIGURE 1.4.10

In both interpenetrated and polycatenated species the individual motifs cannot be separated

"without breaking rings" and, according to the concepts of chemical topology, both classes give

nontrivial entanglements, in the sense that the whole arrays can be considered "topological

isomers" [9,10] of their component motifs (like an n-catenane vs. the separated n rings).

At difference from the above species, other entangled systems, like polyrotaxanes, poly-pseudo-

rotaxanes, interweaved chains and infinite multiple helixes, are all trivial (separable) entanglements

in a strict topological sense. Polyrotaxanes deserve some comments. They are currently described

as interpenetrating systems, i.e. that cannot be disentangled [2], but this relies on metrical

(dimensions of the components, geometrical rigidity) or energy (barriers to dethreading)

considerations rather than on topological ones. By analogy with the topological nomenclature for

stereoisomers [10] we have suggested [3] to call interpenetrated and polycatenated nets

"Topological" entanglements, while polyrotaxane arrays "Euclidean" entanglements (see Figure

1.4.11).

FIGURE 1.4.11

1.4.4.2 Interpenetrating networks

The "topology of interpenetration" is, at present, a still poorly explored field of chemical topology

[59]. Batten and Robson [2] have treated this phenomenon through the analysis of a number of

different real cases, and have introduced a currently accepted nomenclature on this subject.

The classification is based on recognizing the dimensionality (1D, 2D, 3D), the connectivity of the

nodes and the topology of the individual interpenetrating motifs, the degree of interpenetration and

the possible modes of interpenetration. The last point, the less known, is concerned with the relative

disposition of the individual motifs in the three-dimensional space and the consequent mutual

interlacing modes. It is presently difficult to answer to questions like: how many topologically

distinct interpenetration modes are allowed for an n-fold diamondoid network? and also: how can

we recognize when two arrays with the same degree of interpenetration and the same topology of

13

the individual motifs are "topologically identical" or not? We can only state that an acceptable

definition could be the following: two arrays comprised of the same number of interpenetrating

nets with the same topology can be consider "topologically identical" when distorsions can be

performed that bring all the nodes to coincidence without breaking/crossing links (i.e. we could say

that, by analogy with what stated for single nets, they are "ambient isotopic" – see above).

Distinct modes of interpenetration for 2D layers [2] and dia nets [60] have been discussed.

According to the concepts introduced in Figure 1.4.10 we can have three types of interpenetrating

nets based on the dimensionality (1D, 2D and 3D).

Only one case of 1D interpenetration based on hydrogen-bonded ladder motifs has been as yet

reported [61]. On the other hand, a large variety of 2D interpenetrating networks is presently known

(with a maximum of 6-fold interpenetrated 63 hexagonal layers [62]). 3D interpenetrating networks

are quite numerous, with interpenetration degree ranging from 2 to 18. For some times the world

records for the degree of interpenetration belonged to two diamondoid networks: within

coordination polymers to the10-fold interpenetrated [Ag(ddn)2](NO3) (ddn = 1,12-

dodecanedinitrile) [60b], while within hydrogen bonded organic supermolecules to an 11-fold

interpenetrated framework containing molecules of a tetraphenol as tetrahedral centres and

benzoquinone units as rods [63]. Only very recently Zaworotko and coworkers have described the

exceptional structure of the H-bonded net formed by trimesic acid and 1,2-bis(4-pyridyl)ethane in

the ratio 2:3, containing a 18-fold srs net [64].

The analysis and comparison of interpenetrated networks is often a difficult task and a time

expensive work because of the structural complexity of these systems and the enormous growth of

their number, that requires the use of some computer-aided procedure. A recent program suited for

this purpose is TOPOS. The automatic investigation with this package [30] has many advantages: i)

the possibility to process a large number of structural data; ii) the automatic determination of the

interpenetration degree and network relationships; iii) the automatic simplification and topological

classification. This analysis of the crystal structures has evidenced that distinct classes can be

recognized, corresponding to the different modes in which individual identical 3D motifs can

interpenetrate, that are represented by the operations generating the overall array from a single net.

The approach is strictly related to the actual crystallographic structures rather than to the "idealized"

simplified networks in their highest symmetry. Three classes are defined, independently from the

network topology (see Figure 1.4.12) [for the full details on these classes see refs. 14,15].

Class I (Translational). The individual nets are exclusively related by translations. The degree of

interpenetration Z corresponds to the translational degree of interpenetration Zt. There are two

distinct subclasses (Ia and Ib) depending on the presence or not of a full interpenetration vector. In

14

class Ia all the independent nets are related by a single vector and the whole interpenetrated array is

generated by translating a single net (Zt-1) times this vector. Note that many different full

interpenetration vectors can exist in species belonging to this class; of them the shortest one is

selected. On the other hand, in class Ib a full interpenetration vector does not exist and the whole

array is generated by application of more than one translational operation. While class Ia is

numerous the examples of class Ib are much rare (less than 3% of class I).

Class II (Non-translational). The individual nets are related by means of space group symmetry

elements, mainly inversion centres, but also proper rotational axes, screw axes and glide planes. The

degree of interpenetration Z corresponds to the non-translational degree Zn, i.e. the order of the

symmetry element that generates the interpenetrated array from the single net. In almost all cases Zn

is 2, but few examples with Zn up to 4 are known.

The existence of compounds containing more than one interpenetration symmetry element cannot

be ruled out. This require the introduction of an additional sub-classification, similar to the division

of class I, i.e. class IIa with a unique full interpenetration symmetry element and class IIb with

different partial interpenetration symmetry elements (only one real example is known [15]).

Class III (Translational and non-translational). The overall entanglement is generated both by pure

translations and by space group symmetry elements. The value of Z is given by the product Zt x Zn.

FIGURE 1.4.12

The complete search for 3D equivalent interpenetrating networks with TOPOS in the CSD and

ICSD structural databases has produced lists of 301 and 144 reference codes, respectively.

It was observed that 57% of the structures falls in class I of translational interpenetration (that

favours high degrees of interpenetration), while 40% belongs to class II (non-translational) with

degree of interpenetration equal to 2 for almost all (there are only 5 known structures with degree 3

or 4). Class III is rather unusual (3%) and presents always a degree of interpenetration of 4 or

more; the highest known case of 18-fold srs interpenetration quoted above belongs to this class.

It was recently proposed [65] that interpenetrating networks (called "catenated" therein) could be

differentiated on the basis of the relative displacement of the subnets: so we have "interpenetration",

when the frameworks are maximally displaced from each other, or "interweaving" when they are

minimally displaced and exhibit many close contacts. However, a quantitative criterion to establish

when the separation is maximal or minimal is still lacking, and difficult to find especially with a

degree of interpenetration exceeding 2.

15

1.4.4.3 Polycatenated networks

The main feature of polycatenation consists in that the whole catenated array has a higher

dimensionality then that of each of the component motifs. These motifs can be, in principle, 0D, 1D

or 2D species that must contain closed loops and that are interlocked, as for interpenetrating nets,

via topological Hopf links [31] (see Figure 1.4.11). Each motif can be catenated with a finite or also

with an infinite number of other independent motifs but not with all (see Figure 1.4.10).

The classification of these entanglements can be established again by assigning in each case: i) the

dimensionality and topology of the individual motifs; ii) the mode of catenation, i.e. the mutual

orientation and interrelation of the component motifs, and iii) the "degree of entanglement".

The last point requires some comments. While in a n-fold interpenetrated networks we can easily

assign the degree of interpenetration since the whole array contains a finite number of independent

nets, in a polycatenated species we are forced to introduce different concepts since each individual

motif can give finite or infinite interlocking with the other motifs (Figure 1.4.10). In the former case

we must specify only the number n of motifs effectively entangled with each motif, while in the

second case, the number of rings of other motifs interlocked with a single ring of each component

should be specified (and in general different values are possible for the different components).

The different types of polycatenated species can be enumerated and classified on the basis of the

increasing dimensionality of the component motifs (0D, 1D or 2D)[3]. Finite (0D) motifs,

containing closed circuits, can give, in principle, catenation into infinite periodic arrays (from 1D

up to 3D), though no real example has been as yet characterized. Examples of polycatenated species

containing many molecular rings have been described within complex organic species [54], proteins

[66] and synthetic DNA assemblies [67].

Polycatenation of 1D and 2D nets can occur essentially in two modes that are described as

"parallel" and "inclined", according to the nomenclature introduced by Batten and Robson [2].

Different types of 1D motifs are suitable to give polycatenation, including chains of alternating

rings and rods, ribbons of rings, molecular ladders and more complex species. Some examples are

illustrated in Figure 1.4.13. Almost all the known real cases are, at present, based on infinite

molecular ladders interlocked to give 2D or 3D entanglements [68]. An example of catenated

tubular motifs giving 2D layers has also been recently reported [69] (see Figure 1.4.13).

FIGURE 1.4.13

Interlocked 2D layers can exhibit different relative orientations of the planes of the independent

motifs. When all the entangled sheets are parallel and have a common average plane the resulting

16

array is an n-fold interpenetrated 2D structure, with no increase in dimensionality and a finite

number of interweaved motifs (see an example in Figure 1.4.14). If, otherwise, the independent

layers show average planes that are parallel but displaced in a perpendicular direction, they generate

a polycatenated 3D architecture. This "parallel" catenation can occur either because the sheets are

undulated simple layers or because they are multiple layers, in both cases exhibiting some thickness

(see Figures 1.4.14, left, and 1.4.15). Many cases are known that have been previously reviewed

[3].

FIGURE 1.4.14

FIGURE 1.4.15

Polycatenation can also involve different sets of 2D sheets (usually two) that cross at a certain

angle, in the mode that has been called "inclined" [2] (see Figure 1.4.14, right). The examples of

this class are numerous, the majority of which consists of two identical sets of 2D parallel layers, of

63 or 44 topology, spanning two different stacking directions. In these species there is an increase of

dimensionality (2D → 3D) and each individual layer is catenated with an infinite number of other

inclined layers but, obviously, not with all the frames contained in the 3D array. Moreover, in the

analysis of these species different modes of interlocking of the two independent sets have been

envisaged for both topologies [2]. For 44 layers three possible arrangements have been suggested

by Zaworotko and coworkers [70], called parallel-parallel (p-p), parallel-diagonal (p-d) and

diagonal-diagonal (d-d), depending on how the networks orient and penetrate through each other

(see some examples in Figure 1.4.16). Systematic enumerations of these entangled species have

already appeared in some reviews [2,44,48a,56,70].

FIGURE 1.4.16

Only in few cases major variations of the above inclined polycatenation scheme were observed.

For instance, catenation can involve two sets of 2D layers of different topology, as in

[Ni(azpy)2(NO3)2]2[Ni2(azpy)3(NO3)4].4CH2Cl2 [azpy = trans-4,4'-azobis(pyridine)], containing

both 63 and 44 layers [71]. Another variation is represented by the presence of more than two

distinct sets of layers. No real case was known in 1998 when Batten and Robson [2] had suggested

the possibility of finding three different mutually perpendicular stacks; since then few examples

have been discovered and one is illustrated in Figure 1.4.16 (left), namely

[Ni6(bpe)10(H2O)16](SO4)6. xH2O [bpe = 1,2-bis(4-pyridyl)ethane] [72], that contains three distinct

17

sets of 44 layers spanning three different spatial orientations and giving inclined mutual

penetration.

Polycatenation involving motifs of different dimensionality is also known, though quite rare, as in

[Cu5(bpp)8(SO4)4(EtOH)(H2O)5](SO4).EtOH.25.5H2O [bpp = 1,3-bis(4-pyridyl)propane] [73]

containing two different polymeric motifs entangled to give an unique 3D array: ribbons of rings

and two-dimensional tessellated sheets of 44 topology (Figure 1.4.16, right).

1.4.4.4 Borromean networks

Molecular motifs containing different rings could give inextricable entanglements not only via Hopf

links. An alternative way involving at least three closed circuits at a time is represented by the

Borromean link [3,32] (see Figure 1.4.11). This is a nontrivial link in which three rings are

entangled in such a way that any two component rings form a trivial link, i.e. if any one ring is cut

the other two are free to separate. Chemists for a long time have considered the realization of a

molecular Borromean link a synthetic goal of great interest; an example has been very recently

reported by Stoddart and coworkes, comprised of three macrocyclic Borromean rings obtained by

metal templated synthesis [74]. Previously, only one species was known, constructed from single-

stranded DNA by Seeman and coworkers in 1997 [75]. Using Borromean links, however, infinite

1D, 2D or 3D arrays could be imagined. Links like these could appear only mathematical

curiosities; indeed they have been identified in some real examples of infinite interlocked nets.

Borromean links are present in two types of arrays: (2D → 2D) entanglements of three layers and

(2D → 3D) entanglements of infinite layers [32].

Three cases of Borromean 3-fold entangled 2D layers have been recognized, all exhibiting a 63

layer topology [32,76] (see Figure 1.4.11, upper right; one can easily realize that the black net is

completely located above the grey one, the grey net above the white one, but the grey net lies above

the white one).

FIGURE 1.4.17

A Borromean (2D → 3D) entanglement can be envisaged in the structures of two isomorphous

polymeric silver(I) complexes reported by Chen and coworkers, [Ag2(H2L)3](NO3)2 and

[Ag2(H2L)3](ClO4)2 [H2L = N,N'-bis(salicylidene)-1,4-di-aminobutane] [77] and in another silver

species with a similar formula [78]. They are comprised of highly undulated 63 layers of 3-

connected silver centres (Figure 1.4.17). The silver atoms exhibit interlayer unsupported Ag…Ag

interactions and, taking into account these contacts, a single 4-connected 3D array results, that can

be described as self-catenated embedding of dia (regular dia - see e.g. Figures 1.4.5, 1.4.7 and 1.4.8

18

- and self-dia are non-"ambient isotopic" and have obviously the same Schläfli Symbol, Vertex

Symbol, as well as the Coordination Sequence, but dia cannot be deformed into self-dia without

edge crossings) [32,34]. On the other hand, and much more interestingly, if the Ag…Ag

interactions are neglected, the layers are not (2D → 3D) polycatenated but are interlinked via

Borromean links involving three layers at a time. The whole array represents the first infinite case

of n-Borromean links (Figure 1.4.17).

1.4.4.5 Other entanglements

Other known types of infinite entanglements are comprised of finite or infinite components that can

be "ideally" separated without breaking effective topological links. We are speaking of

polythreaded arrays, i.e. extended periodic analogues of molecular rotaxanes and pseudo-rotaxanes.

The term "Euclidean entanglements" was suggested for polyrotaxanes, meaning that only the

presence of geometrical or energetical (non topological) constrains can prevent the separation of

the individual motifs [79]. On the other hand, poly-pseudo-rotaxanes contain motifs that are

threaded by chains or strings which, in principle, can be slipped off; they display a type of

entanglement that is neither topological nor Euclidean.

Polythreaded systems contain motifs that are interweaved via rotaxane-like mechanical links and

this implies the presence of closed loops as well as of elements that can thread the loops (Figure

1.4.2). These two types of moieties may belong to the same unit or may be separately supplied by

motifs having different structures. The constituent motifs could be, in principle, 0D species, 1D

polymers or arrays of higher dimensionality. The resulting array can show the same or an increased

dimensionality with respect to that of the polythreaded units.

Known examples involving 0D motifs, i.e. molecular "beads" (cucurbituril molecules), threaded by

coordination polymers of different topology have been described by Kim and coworkers [80].

Many examples of extended systems that "cannot be disentangled" have been reported very

recently. The 1D chain motif comprised of alternating rings and rods is particularly suitable for

polythreading since it contains both the elements needed in a rotaxane-like mechanical linkage.

Chains of this type have been found to give (1D → 1D) [81], (1D → 2D) parallel [82] (see Figure

1.4.18, a and b) and inclined [82, 83] (see Figure 1.4.11, bottom right) polythreaded arrays.

FIGURE 1.4.18

19

Numerous entangled arrays containing "separable" motifs (poly-pseudo-rotaxanes), like those

shown in Figure 1.4.18 (right) have also been reported [see ref. 3].

Architectures of interweaved 1D motifs display different types of entanglements that are neither

topological nor Euclidean. Polymeric chains, both rigid or flexible, represent the most simple and

common structural extended motif. The packing of 1D polymers usually occurs with parallel

orientation of all the chains; less commonly they can span two different directions on alternate

layers. Independent chains can be sometimes connected into pairs or into extended 2D or 3D arrays

by weak supramolecular interactions. Quite rare are, on the other hand, other types of associations

of the chains resulting in entangled systems of unusual topologies, including chains woven like

warp-and-woof threads in a cloth to give 2D sheets (Figure 1.4.19, left) and infinite multiple helixes

(Figure 1.4.19, right).

FIGURE 1.4.19

1.4.5. Conclusions

We have attempted to rationalize here the structures of extended frames at different levels of

complexity in terms of the network topology. With the aim to establish useful relationships between

structures and properties [84] a careful analysis of the topology is unavoidable, particularly for the

complicated species that are more and more discovered. Indeed, phenomena like interpenetration

are not only structural curiosities, with some esthetical appeal, but can play an effective role on the

control of the properties of materials, as shown, inter alia, by their influence on network porosity

[55a,65a] and by their capability to produce anomalous magnetic properties [85].

20

References

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21

Kristallogr. 2004, 219, 528; R. T. Strong, C. J Pickard, V. Milman, G. Thimm, B. Winkler, Phys. Rev. 2004, B70, 45101. [21] O. Delgado-Friedrichs, M. O'Keeffe, J. Solid State Chem. 2005, 178, 2480. [22] This nomenclature includes some simply related nets as the augmented net and the edge net called xxx-a and xxx-e respectively. An augmented net is one in which the vertices of the original net are replaced by a polygon or polyhedron corresponding to the original coordination figure. The edge net (also expanded net) is obtained from the original net by placing vertices in the middle of the edges, discarding the original vertices and edges, and joining the new vertices to enclose the coordination figure of the original vertices. Moreover decorated nets (xxx-f,g etc.) are obtained by replacing the vertices of the original net by groups of vertices. Augmented nets are special cases of decorated nets. See RCSR web page and ref. 23. [23] (a ) O. Delgado Friedrichs, M. O’Keeffe, O.M. Yaghi, Acta Crystallogr. 2003, A59, 22; (b) O. Delgado Friedrichs, M. O’Keeffe, O.M. Yaghi, Acta Crystallogr. 2003, A59, 515. [24] (a) C. Baerlocher, W.M. Meier, D.H. Olsen, Atlas of Zeolite Framework Types, Elsevier, Amsterdam, 2001; (b) http://www.iza-structure.Org/databases/ ; (c) M. M. J. Treacy, Microporous Mesoporous Mater. 2003, 58, 1. [25] N. W. Ockwig, O. Delgado-Friedrichs, M. O'Keeffe and O. M. Yaghi, Acc. Chem. Res. 2005, 38, 176. [26] In a more rigorous graph-theoretical approach we should use the term n-coordinated instead of n-connected, but we keep here the word connected to indicated what is clear to chemists: for example, a metal could be 6-coordinated but give only a 3-connected net if three coordination sites are occupied by ligands that are not links of the network. [27] M. O’Keeffe, B.G. Hyde, Phil. Trans. R. Soc. London, 1980, 295A, 553. [28] An equivalent definition for ring is an n-membered circuit that represents the shortest possible path connecting all the [n(n-1)/2] pairs of nodes belonging to that circuit. [29] M. O’Keeffe, S.T. Hyde, Zeolites 1977, 19, 370. [30] The program package TOPOS 4.0 with an advanced graphical interface (available at http://www.topos.ssu.samara.ru) was recently improved for the automatic determination and classification of the interpenetration degree, see refs. 14 and 15. The program is also able to analyze and simplify complex groups (including H-bonded networks) and to assign the topology of the resulting network according to the RCSR proposed symbol. As a result, typical crystal structure of organic, inorganic, or coordination compound may be processed in a few minutes from the .res or .cif files, to get a comprehensive description of the topology and of the interpenetration (if any). A detailed analysis of Voronoi-Dirichlet partition of crystal space and estimation of related crystallochemical parameters are also available in TOPOS. See, V.A. Blatov, Cryst. Rev. 2004, 10, 249. [31] E. Flapan, When Topology Meets Chemistry, Cambridge University Press, 2000. [32] L. Carlucci, G. Ciani, D.M. Proserpio, CrystEngComm 2003, 5, 269. [33] E. Koch and H. Sowa, Acta Crystallogr. 2004, A60, 239, W. Fischer, Acta Crystallogr. 2004, A60, 246. [34] O. Delgado-Friedrichs, M. D. Foster, M. O'Keeffe, D. M. Proserpio, M. M. J. Treacy, O. M. Yaghi, J. Solid State Chem. 2005, 178, 2533. [35] (a) O. Delgado-Friedrichs, M. O’Keeffe, Acta Crystallogr. 2003, A59, 351; (b) O. Delgado-Friedrichs, Nova Acta Leopold. NF, 2003, 88, 39; (c) O. Delgado-Friedrichs, Lecture Notes in Computer Science, vol. 2912, Springer, Berlin, 2004, 178; (d) O. Delgado-Friedrichs, Discrete Comput. Geom. 2005, 33, 67. [36] (a) R. Robson, J. Chem. Soc., Dalton Trans. 2000, 3735; (b) O. M. Yaghi, M. O'Keeffe, N. W. Ockwig, H. K. Chae, M. Eddaoudi, J. Kim, Nature 2003, 423, 705. [37] G. Férey, J. Solid State Chem. 2000, 152, 37. [38] (a) O. Delgado-Friedrichs, D. H. Huson, Discr. Comput. Geom. 1999, 21, 229; (b) O. Delgado-Friedrichs, A. W. M. Dress, D. H. Huson, J. Klinowsky, A. L. Mackay, Nature 1999, 400, 644; (c)

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M. O'Keeffe, Nature, 1999, 400, 617; (d) O. Delgado-Friedrichs, D. H. Huson, Discr. Comput. Geom. 2000, 24, 279. [39] (a) O. Delgado Friedrichs, M. O’Keeffe, O.M. Yaghi, Solid State Sci. 2003, 5, 73; (b) O. Delgado Friedrichs, M. O’Keeffe, Acta Crystallogr. 2005, A61, 358. [40] Another application of the tiling theory is in the search of new zeolites, see the most recent results in: (a) M.D. Foster, A. Simperler, R.G. Bell, O. Delgado Friedrichs, F. A. Almeida Paz, J. Klinowski, Nature Mater. 2004, 3, 234; (b) A. Simperler, M.D. Foster, O. Delgado Friedrichs, R.G. Bell, F.A. Almeida Paz, J. Klinowski, Acta Crystallogr. 2005, B61, 263. [41] (a) S. Andersson, S. T. Hyde, H. G. von Schnering, Z. Kristallogr. 1984, 168, 1; (b) H. G. von Schnering, R. Nesper, Angew. Chem. Int. Ed. Engl. 1987, 26, 1059; (c) S. Andersson, S. T. Hyde, K. Larsson, S. Lidin, Chem. Rev. 1988, 88, 221; (d) S. T. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B. W. Ninham, The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry, and Biology, Elsevier, Amsterdam, 1997; R. Nesper, S. Leoni, ChemPhysChem 2001, 2, 413. [42] (a) A. H. Schoen, NASA Techn. Note, 1970, D-5541; (b) W. Fischer, E. Koch, Phil. Trans. R. Soc. Lond. 1996, A354, 2105; (c) C. Bonneau, O. Delgado Friedrichs, M. O’Keeffe, O.M. Yaghi, Acta Crystallogr. 2004, A60, 517. [43] (a) V. Robins, S.J. Ramsden, S.T. Hyde, Eur. Phys. J. B, 2004, 39, 365; (b) S. T. Hyde, A.-K. Larssen, T. Di Matteo, S. Ramsden, S. T. Hyde, Aust. J. Chem. 2003, 56, 981. [44] S.R. Batten, R. Robson, in: J.-P. Sauvage, C. Dietrich-Buchecker (Eds.), Molecular Catenanes, Rotaxanes and Knots: A Journey Through the World of Molecular Topology, Wiley-VCH, Weinheim, 1999, pp. 77–105. [45] M.A. Withersby, A.J. Blake, N.R. Champness, P.A. Cooke, P. Hubberstey, M. Schröder, J. Am. Chem. Soc. 2000, 122, 4044. [46] D.J. Price, S.R. Batten, B. Moubaraki, K.S. Murray, Chem. Eur. J. 2000, 6, 3186. [47] M. O’Keeffe, Z. Kristallogr. 1991, 196, 21. [48] (a) S.R Batten, CrystEngComm. 2001, 3, 67; (b) M. O'Keeffe, Nature 1998, 392, 879. However, taking into account our definition of self-catenation, for icf the catenated six and ten membered circuits are not both "shortest rings"; and for itv the catenated 8-rings are not "strong rings". It follows that a more detailed classification is possible considering also not only the "shortest rings" defining the VS but also larger ring that may exists (as for icf), and the "strong rings" (as for itv) [see ref. 21 for definition of all kind of rings]. [49] B.F. Abrahams, M.J. Hardie, B.F. Hoskins, R. Robson, E.E. Sutherland, J. Chem. Soc. Chem. Commun. 1994, 1049. [50] The same rob topology, together with a new self-catenated 6-connected net, has been recently reported for [Sc(4,4’-bipyridine-N,N’-dioxide)3(ClO4)3]: D.-L. Long, R.J. Hill, A.J. Blake, N.R. Champness, P. Hubberstey, C. Wilson, M. Schröder, Chem. Eur. J. 2005, 11, 1384. [51] B.F. Abrahamas, S.R. Batten, M.J. Grannas, H. Hamit, B.F. Hoskins, R. Robson, Angew. Chem., In. Ed., 1999, 38, 1475. [52] L. Carlucci, G. Ciani, D.M. Proserpio, S. Rizzato, J. Chem. Soc., Dalton Trans. 2000, 3821. [53] Self-catenation is not observed only in 3D nets but is possible also in 2D nets: few rare examples are known [see ref. 3] and two recent are: the 4-connected layers in a Cd coordination polymer (G. O. Lloyd, J. L. Atwood, L. J. Barbour, Chem. Commun. 2005, 1845 and the 6-connected layers in a supramolecular arrangement of a tetrazole derivative (A. T. Rizk, C.A. Kilner, M.A. Halcrow CrystEngComm, 2005, 7, 359. [54] (a) J.-P. Sauvage, Acc. Chem. Res. 1998, 31, 611; (b) J.C. Chambron, C.O. Dietrich-Buchecker, J.-P. Sauvage, Top. Curr. Chem. 1993, 165, 131; (c) J.-P. Sauvage, Acc. Chem. Res. 1990, 23, 319; (d) J.-P. Sauvage, C.O. Dietrich-Buchecker, Molecular Catenanes, Rotaxanes and Knots, Wiley-VCH, Weinheim, 1999; (e) C.O. Dietrich-Buchecker, J.-P. Sauvage, Angew. Chem. Int. Ed. Engl. 1989, 28, 189; (f) D.B. Amabilino, J.F. Stoddart, Chem. Rev. 1995, 95, 2725; (g) F.M.

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Raymo, J.F. Stoddart, Chem. Rev. 1999, 99, 1643; (h) M.C.T. Fyfe, J.F. Stoddart, Acc. Chem. Res. 1997, 30, 393; (i) S.A. Nepogodiev, J.F. Stoddart, Chem. Rev. 1998, 98, 1959. [55] (a) T.M. Reineke, M. Eddaoudi, D.M. Moler, M. O’Keeffe, O.M. Yaghi, J. Am. Chem. Soc. 2000, 122, 4843; (b) M. Kondo, M. Shimamura, S.-I. Noro, S. Minakoshi, A. Asami, K. Seki, S. Kitagawa, Chem. Mater. 2000, 12, 1288. [56] S.R. Batten, Curr. Opin. Solid State. Mater. Sci. 2001, 5, 107. [57] (a) J.Y. Lu, A.M. Babb, Chem. Commun. 2001, 821; (b) Y.-C. Jiang, Y.-C. Lai, S.-L. Wang, K.-H. Lii, Inorg. Chem. 2001, 40, 5320; (c) D.M. Shin, I.S. Lee, Y.K. Cheung, M.S. Lah, Chem. Commun. 2003, 1036. [58] L. Carlucci, G. Ciani, D. M. Proserpio, Chem. Commun. 2004, 380. [59] Two-fold interpenetration in some prototypical nets has been discussed in terms of dual of the tiling of the net by O’Keeffe and coworkers (see ref. 39a). The dual of a tiling is obtained by putting new vertices in the center of the tiles and connecting such vertices by edges passing through common faces (each face of a tiling is common to two tiles) forming an interpenetrating net, so that all rings (faces) of the original structure are linked (catenated) to faces of the dual one and vice versa. When the tiling of the net and its dual are the same, we speak of self-dual nets (e.g. dia, pcu, srs, fcu, cds). [60] (a) L. Carlucci, G. Ciani, P. Macchi, D.M. Proserpio, S. Rizzato, Chem. Eur. J. 1999, 5, 237; (b) L. Carlucci, G. Ciani, D.M. Proserpio, S. Rizzato, Chem. Eur. J. 2002, 8, 1520. [61] H.-F. Zhu, J. Fan, T. Okamura, W.-Y. Sun, N. Ueyama, Chem. Lett. 2002, 898. [62] D. Venkataraman, S. Lee, J.S. Moore, P. Zhang, K.A. Hirsch, G.B. Gardner, A.C. Covey, C.L. Prentice, Chem. Mater. 1996, 8, 2030. [63] D.S. Reddy, T. Dewa, K. Endo, Y. Aoyama, Angew. Chem. Int.Ed. Engl. 2000, 39, 4266. [64] T. R. Shattock, P. Vishweshwar, Z. Wang, M. J. Zaworotko, Cryst. Growth Des. 2005, 5, 2046. [65] (a) B. Chen, M. Eddaoudi, S. T. Hyde, M. O'Keeffe, O. M. Yaghi, Science 2001, 291, 1021; (b) J. L. C. Rowsell, O. M. Yaghi, Angew. Chem. Int. Ed. 2005, 44, 4670. [66] See, eg.: W.R. Wikoff, L. Liljas, R.L. Duda, H. Tsuruta, R.W. Hendrix, J.E. Johnson, Science 2000, 289, 2129; J. Chen, C.A. Rauch, J.A. White, P.T. Englund, N.R. Cozzarelli, Cell 1995, 80, 61. [67] N. C. Seeman, P. S. Lukeman, Rep. Prog. Phys. 2005, 68, 237. [68] See eg.: A. J. Blake, N. R. Champness, A. Khlobystov, D. A. Lemenovskii, W.-S. Li, M. Schröder, Chem. Commun. 1997, 2027; L. Carlucci, G. Ciani, D.M. Proserpio, J. Chem. Soc., Dalton Trans. 1999, 1799. [69] X.-L. Wang, C. Qin, E.-B. Wang, Y.-G. Li, Z.-M. Su, L. Xu, L. Carlucci, Angew. Chem. Int. Ed. Engl. 2005, 44, 5824. [70] K. Biradha, A. Mondal, B. Moulton, M.J. Zaworotko, J. Chem. Soc.,Dalton Trans. 2000, 3837. [71] L. Carlucci, G. Ciani, D.M. Proserpio, New J. Chem. 1998, 1319. [72] L. Carlucci, G. Ciani, D.M. Proserpio, S. Rizzato, CrystEngComm. 2003, 5, 190. [73] L. Carlucci, G.Ciani M. Moret, D.M. Proserpio, S. Rizzato, Angew. Chem. Int. Ed. Engl. 2000, 39, 1506. [74] (a) S. J. Cantrill, K. S. Chichak, A. J. Peters, J. F. Stoddart, Acc. Chem. Res. 2005, 38, 1; (b) K. S. Chichak, S.J. Cantrill, A.R. Pease, S.-H. Chiu, G.W. Cave, J. L. Atwood, J. F. Stoddart, Science 2004, 304, 1308. [75] C. Mao, W. Sun, N. C. Seeman, Nature 1997, 386, 137. [76] (a) D.B. Leznoff, B.-Y. Xue, R.J. Batchelor, F.W.B. Einstein, B.O. Patrick, Inorg. Chem. 2001, 40, 6026 (b) M.P. Suh, H.J. Choi, S.M. So, B.M. Kim, Inorg. Chem. 2003, 42, 676; (c) R. Liantonio, P. Metrangolo, T. Pilati, G. Resnati, Cryst. Growth Des. 2003, 3, 355. [77] M.L. Tong, X.-M. Chen, B.-H. Ye, L.-N. Ji, Angew. Chem. Int. Ed. Engl. 1999, 38, 2237. [78] S. Muthu, J.H.K. Yip, J.J. Vittal, J. Chem. Soc., Dalton Trans. 2002, 4561. [79] Not only poly-pseudo-rotaxanes but also polyrotaxanes should be considered trivial topological entanglements, extending the concepts developed for the related molecular entities. By

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means of "ideal" continuous deformations we could separate the components of any finite portion of a polyrotaxane. However, the extended motifs under examination can be considered infinite entities. Including this additional "boundary condition" a different topological description could result for such interweaving systems, but this is at present an almost completely unexplored area of chemical topology. [80] J. W. Lee, S. Samal, N. Selvapalam, H.-J. Kim, K. Kim, Acc. Chem. Res. 2003, 36, 621. [81] (a) C.J. Kuehl, F.M. Tabellion, A.M. Arif, P.J. Stang, Organomet. 2001, 20, 1956; (b) C.S.A. Fraser, M.C. Jennings, R.J. Puddephatt, Chem. Commun. 2001, 1310. [82] L. Carlucci, G. Ciani, D. M. Proserpio, Cryst. Growth Des. 2005, 5, 37. [83] B. F. Hoskins, R. Robson, D. A. Slizys, J. Am. Chem.Soc. 1997, 119, 2952. [84] (a) C. Janiak, Dalton Trans. 2003 2781; (b) J.S. Miller, Adv. Mater. 2001, 13, 525. [85] J. S. Miller, T. E. Vos, W. W. Shum, Adv. Mater. 2005, 17, 2251.

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Figure Captions Figure 1.4.1. An alternative selection of nodes gives two distinct topological descriptions, here illustrated for the 2-fold interpenetrated H-bonded structure of H2Te2O3F4 [J. C. Jumas, M. Maurin, E. Philippot, J. Fluor. Chem., 1976, 8, 329]: 3-connected ThSi2 (ths) with Te, and diamond (dia) with the molecule centroid. Figure 1.4.2. Models of links and knots for finite entities. Figure 1.4.3. Nets with common topologies and their nomenclature (see text). The symbols are related to prototypical structures or lattices as follows: SrSi2 (srs), NbO (nbo), primitive cubic (pcu), body centered cubic (bcu), face centered cubic (fcu), sodalite (sod), ThSi2 (ths), CdSO4 (cds), quartz (qtz), SrAl2 (sra), PtS (pts), rutile (rtl). Figure 1.4.4 In the 3-connected uninodal net rhr-a the vertex 1 with the three edges a,b,c has Vertex Symbol 4.8.12 (angles ac: 4-ring, ab: 8-ring , bc: 12-ring, with single multiplicity for all three rings: 41.81.121) . If we consider the shortest circuits we get the Schläfli Symbol 4.8.10: infact, the angle bc is included in the circuit with 10 edges [1,2,3,4,5,6,7,8,9,10] that is not a ring because is the sum of two shortest rings (there is a short-cut represented by edge a). Figure 1.4.5 The structures of diamond dia (left) and lonsdaleite (hexagonal diamond) lon (right). The two nets have the same Schläfli and Vertex Symbol VS (top), but different Coordination Sequence (CS). The rings multiplicity for one selected angle is evidenced in grey; in both nets for each angle there are two 6-rings, in dia both have the same chair conformation, in lon the conformation is different, one chair and one boat. Figure 1.4.6 The highly distorted diamondoid net observed in the 2-fold interpenetrated structure of SbCl3(p-diacetylbenzene) with a diamondoid cage in black [W. A. Baker, D. E. Williams, Acta Crystallogr., 1978, B34, 1111] (left). The observed diamondoid cage (center) is compared with the undistorted adamantane cage as in diamond (right). Figure 1.4.7. Two different layered nets (2-periodic) characterized by the same VS and CS that are not ambient isotopic. We cannot transform one into the other without bond/edge crossing. The layer on the left is rather common while that on the right is observed only in [Ag2L2(µ-PO2F2)](PF6) [L = 1-(isocyanidomethyl)-1H-benzotriazole] [I. Ino, J.C. Zhong, M. Munakata, T. Kuroda-Sowa, M. Maekawa, Y. Suenaga, Y. Kitamori, Inorg. Chem. 2000, 39, 4273]. Figure 1.4.8. The structures of diamond (dia) (left) and lonsdaleite (hexagonal diamond, lon) (right) showing in grey the natural tile with 10 vertices for dia (adamantane cage) with transitivity [1111] and the two natural tiles with 8 and 12 vertices for lon with transitivity [1222]. Figure 1.4.9. Three examples of self-catenated nets with two catenated rings (Hopf links) in black: 6-rings (rob), 8-rings (coe), 12-rings (twt). Figure 1.4.10. The differences of interpenetration vs. polycatenation. Figure 1.4.11. The overall view of the entanglement phenomena in extended compounds (coordination and H-bonded). Figure 1.4.12. The different classes of interpenetration (translational and non-translational) illustrated with an example of 3-connected 2-fold ths.

26

Figure 1.4.13. Polycatenation of 1D motifs: 2D entangled layers from parallel catenation of undulating ladders (laterally catenated by two other ladders by each side) (left), and of nanotube motifs (center); 3D superstructure from catenation of ladders running in different directions (right). Figure 1.4.14. Three possible topological entanglements of hexagonal 63 layers showing the interpenetration versus catenation (parallel and inclined). Figure 1.4.15. The 3D entanglement of parallely catenated double layers (left) and an intricate case of parallel catenation of undulated 2-fold 63 layers (right) [S. Banfi, L. Carlucci, E. Caruso, G. Ciani, and D.M. Proserpio, Cryst. Growth Des. 2004, 4, 29] Figure 1.4.16. The 3D entanglements of different inclined catenations: three sets of 2D (44) layers(left), perpendicular catenation of 2-fold 44 layers (center) [K.A. Hirsch, S.R. Wilson, J.S. Moore, Chem. Commun. 1998, 13], 44 layers catenated by perpendicular 1D ribbons of rings (right). Figure 1.4.17. A Borromean (2D → 3D) entanglement of undulated (63) layers (left and center, see refs. 32,77,78). On the right is illustrated the related self-dia net discussed in the text (see also refs. 32,34). Figure 1.4.18. Examples of polyrotaxanes and poly-pseudo-rotaxanes. Figure 1.4.19. A warp-and-woof like 2D layer of 2-over/2-under chains (left) [Y.-H. Li, C.-Y. Su, A.M. Goforth, K.D. Shimizu, K.D. Gray, M.D. Smith, H.-C. zur Loye, Chem. Commun. 2003, 1630] and interlaced 5-fold helixes (right) [Y. Cui, S.J. Lee, W. Lin, J. Am. Chem. Soc. 2003, 125, 6014].

27

Figure 1.4.1. An alternative selection of nodes gives two distinct topological descriptions, here illustrated for the 2-fold interpenetrated H-bonded structure of H2Te2O3F4 [J. C. Jumas, M. Maurin, E. Philippot, J. Fluor. Chem., 1976, 8, 329]: 3-connected α-ThSi2 (ths) with Te, and diamond (dia) with the molecule centroid.

Figure 1.4.2. Models of links and knots for finite entities.

Figure 1.4.3. Nets with common topologies and their nomenclature (see text). The symbols are related to prototypical structures or lattices as follows: SrSi2 (srs), NbO (nbo), primitive cubic (pcu), body centered cubic (bcu), face centered cubic (fcu), sodalite (sod), ThSi2 (ths), CdSO4 (cds), quartz (qtz), SrAl2 (sra), PtS (pts), rutile (rtl).

1

Figure 1.4.4 In the 3-connected uninodal net rhr-a the vertex 1 with the three edges a,b,c has Vertex Symbol 4.8.12 (angles ac: 4-ring, ab: 8-ring , bc: 12-ring, with single multiplicity for all three rings: 41.81.121) . If we consider the shortest circuits we get the Schläfli Symbol 4.8.10: infact, the angle bc is included in the circuit with 10 edges [1,2,3,4,5,6,7,8,9,10] that is not a ring because is the sum of two shortest rings (there is a short-cut represented by edge a).

Figure 1.4.5 The structures of diamond dia (left) and lonsdaleite (hexagonal diamond) lon (right). The two nets have the same Schläfli and vertex symbol VS (top), but different coordination sequence (CS). The rings multiplicity for one selected angle is evidenced in grey; in both nets for each angle there are two 6-rings, in dia both have the same chair conformation, in lon the conformation is different, one chair and one boat.

Figure 1.4.6 The highly distorted diamondoid net observed in the 2-fold interpenetrated structure of SbCl3(p-diacetylbenzene) with a diamondoid cage in black [W. A. Baker, D. E. Williams, Acta Crystallogr. Sect. B, 1978, 34, 1111] (left). The observed diamondoid cage (center) is compared with the undistorted adamantane cage as in diamond (right).

2

Figure 1.4.7. Two different layered nets (2-periodic) characterized by the same VS and CS that are not ambient isotopic. We cannot transform one into the other without bond/edge crossing. The layer on the left is rather common while that on the right is observed only in [Ag2L2(µ-PO2F2)](PF6) [L = 1-(isocyanidomethyl)-1H-benzotriazole] [I. Ino, J.C. Zhong, M. Munakata, T. Kuroda-Sowa, M. Maekawa, Y. Suenaga, Y. Kitamori, Inorg. Chem. 2000, 39, 4273].

Figure 1.4.8. The structures of diamond (dia) (left) and lonsdaleite (hexagonal diamond, lon) (right) showing in grey the natural tile with 10 vertices for dia (adamantane cage) with transitivity [1111] and the two natural tiles with 8 and 12 vertices for lon with transitivity [1222].

Figure 1.4.9. Three examples of self-catenated nets with two catenated rings (Hopf links) in black: 6-rings (rob), 8-rings (coe), 12-rings (twt).

Figure 1.4.10. The differences of interpenetration vs. polycatenation.

3

Figure 1.4.11. The overall view of the entanglement phenomena in extended compounds (coordination and H-bonded).

Figure 1.4.12. The different classes of interpenetration (translational and non-translational) illustrated with an example of 3-connected 2-fold ths.

Figure 1.4.13. Polycatenation of 1D motifs: 2D entangled layers from parallel catenation of undulating ladders (laterally catenated by two other ladders by each side) (left), and of nanotube motifs (center); 3D superstructure from catenation of ladders running in different directions (right).

Figure 1.4.14. Three possible topological entanglements of hexagonal 63 layers showing the interpenetration versus catenation (parallel and inclined).

4

Figure 1.4.15. The 3D entanglement of parallely catenated double layers (left) and an intricate case of parallel catenation of undulated 2-fold 63 layers (right) [S. Banfi, L. Carlucci, E. Caruso, G. Ciani, D.M. Proserpio, Cryst. Growth Des. 2004, 4, 29]

Figure 1.4.16. The 3D entanglements of different inclined catenations: three sets of 2D 44 layers (left), perpendicular catenation of 2-fold 44 layers (center) [K.A. Hirsch, S.R. Wilson, J.S. Moore, Chem. Commun. 1998, 13], 44 layers catenated by perpendicular 1D ribbons of rings (right).

Figure 1.4.17. A Borromean (2D → 3D) entanglement of undulated 63 layers (left and center, see refs. 32,77,78). On the right is illustrated the related self-dia net discussed in the text (see also refs. 32,34).

Figure 1.4.18. Examples of polyrotanes and polypseudo-rotaxanes.

Figure 1.4.19. A warp-and-woof like 2D layer of 2-over/2-under chains (left) [Y.-H. Li, C.-Y. Su, A.M. Goforth, K.D. Shimizu, K.D. Gray, M.D. Smith, H.-C. zur Loye, Chem. Commun. 2003, 1630] and interlaced 5-fold helixes (right) [Y. Cui, S.J. Lee, W. Lin, J. Am. Chem. Soc. 2003, 125, 6014].

5